Articles of factoring

How to factor $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$?

The term is: $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$ And the answer is : $(a+b+c)(b+c-a)(c+a-b)(a+b-c)$ I have tried a lot, but could’t accomplish. Please don’t bring up any complex method, it is just a high school math problem. But in vain I just can’t do it.

Is there a branch of mathematics that studies the factors of rational numbers?

Is there a branch of mathematics that studies the factors of rational numbers? I am imagining that defining this would work pretty much the same way as defining the factors x of an integer n: $\{x \mid n\mod x\ = 0\}$ but maybe as something more like $\{x^{-1} \mid n\mod x\ = 0\}$ giving the […]

Computing square roots implies factoring $n = pq$

I’m proving that computing square roots in $\mathbb{Z}_{pq}$ implies factoring $n = pq$ with $p,q$ primes. The solution give you an algorithm: repeat 1. pick y from {1,…,n-1} 2. x = y^2 mod n 3. y’ = random square root of x until y’ != y and y’ != – y mod n Basically the […]

irreducibility of polynomials with integer coefficients

Consider the polynomial $$p(x)=x^9+18x^8+132x^7+501x^6+1011x^5+933x^4+269x^3+906x^2+2529x+1733$$ Is there a way to prove irreducubility of $p(x)$ in $\mathbb{Q}[x]$ different from asking to PARI/GP?

Factoring extremely large integers.

The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so would appreciate a verbose description of equations. The Problem: The integer in question will ALWAYS be a power of […]

How to solve Diophantine equations of the form $Axy + Bx + Cy + D = N$?

Is there a general solution to solve a Diophantine equation of the form $Axy + Bx + Cy + D = N$? With $A,B,C,D,N,x,y$ positive integers.

How to factorize polynomial in GF(2)?

I want to know how to factorize polynomials in $\ GF(2) $ without a calculator in a product of irreducible factors. For example, in my exercises, I must factorize $\ p(x) = (x^7 – 1)$ The response is $\ (x − 1)(x^3 + x^2 + 1)(x^3 + x + 1)$ I just not understand the […]

When can a function not be factored?

Are there any general conditions under which a function involving $n$ unknowns cannot be factored into a product of $n$ terms each of which contains only one of the unknowns? For example, $xy$ can be factored into $x$ and $y$ but we cannot factor $1/(1+xy)$ into a product of terms, each of which only contains […]

Factorise $y^2 -3yz -10z^2$

How do I solve this question? I have looked at the problem several times. However, I cannot find a viable solution. I believe that it is a perfect square trinomial problem.

Number of Relatively Prime Factors

Given a number $n$, in how many ways can you choose two factors that are relatively prime to each other (that is, their greatest common divisor is 1)? Also, am I going in the correct direction by saying if $n$ is written as $p_1^{a_1}p_2^{a_2}\dots p_k^{a_k}$, where $p_i$ is a prime and $a_i\geq 1$, then the […]